On the suspension of implicatures
Anton Benz
Centre for General Linguistics, Berlin
Abstract
In this talk we apply the idea of preferential models from nonmonotonic semantics to game theoretic pragmatics in order to explain the mechanisms underlying
the cancellation and suspension of implicatures. Preferential models represent assumptions about what is normal. We use them to represent assumptions about what
are normal utterance situations. If the use of a form contradicts these assumptions
about normality, then the addressee can conclude that the utterance situation is
non-normal. This then can lead to the suspension of implicatures.
1
Introduction
The topic of this paper is an old one: that of suspension and cancellation of implicatures.
The difference between these two forms of annulling an implicatures can be seen from the
following examples:
(1)
a. Cancellation: Some, in fact all, of the boys came to the party.
b. Suspension: Some, perhaps all, of the boys came to the party.
In both cases, the scalar implicature from some to some but not all is cancelled. In case of
cancellation proper, there is an explicit contradiction between the scalar implicature and
the literal content of the utterance. In the case of suspension, the speaker only asserts
the possibility of its negation (Horn 1989: pp. 234-235). The classical account is that of
Gazdar (1979) who assumes that cancellation and suspension result from an incremental
amplification of utterance interpretation. In this amplification, firstly all logical consequences are added to an utterance, secondly, if consistent, all clausal implicatures, thirdly
all scalar implicatures, and finally all presuppositions. Hence, annulation of scalar implicatures is explained by a contradiction with the logical consequences, and suspension by
a contradiction with the clausal implicatures of an utterance.
In this paper, we propose a game theoretic model which not only explains the cancellation and suspension of scalar implicatures but also of relevance implicatures. We discuss
the following principled examples, which cannot be explained by a Gazdarian incremental
account:
(2)
a. A: Does this job candidate speak Spanish?
i. He speaks Portuguese. +> He does not speak Spanish.
ii. B: I know he speaks Portuguese. +> B does not know whether he speaks
Spanish.
b. A: How did the students do in the exam?
i. B: Some students passed. +> Not many passed.
ii. B: I know that some students passed. +> B does not know whether many
passed.
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73
The first example, (2ai), shows a relevance implicature. Hirschberg (1991) explained it
as a generalised scalar implicature which results from a relevance scale. The suspensions
in (2aii) and (2bii) cannot be explained by an incompatibility with a clausal implicature
as know does not generate clausal implicatures (Gazdar 1979; Levinson 2000). Hence, an
incremental account would predict that the (i) and (ii) versions lead to exactly the same
implicatures.
From a logical point of view, implicatures are a kind of nonmonotonic inferences. A
large class of nonmonotonic logics are concerned with inferences to what normally holds.
For these logics, normality can be semantically defined by preferential models (Shoham
1987; Schlechta 2004). We borrow the idea of preferential models and apply it to a game
theoretic model of implicatures as developed in Benz & van Rooij (2007).
Speaking very generally, a nonmonotonic inference T |∼ R is valid in a nonmonotonic
logic based on preferential models iff M (T ) |= R holds classically in a set M (T ) of preferred
T models. M (T ) represents the assumptions about what normally holds if T is true. Let
us again consider the standard example of a scalar implicature: Some of the boys came to
the party (T ) +> Not all of the boys came to the party (R). In the common explanation
of this implicature, it must be assumed that the speaker knows how many of the boys
came, i.e., M (T ) only contains possible worlds in which the speaker knows the actual
world. If the expert condition is violated, then the implicature doesn’t arise anymore,
as can be seen from (1b). Accordingly, we will propose a game theoretic model in which
a joint preference relation over a partition of all speaker types is defined such that the
implicature of an utterance is calculated with respect to the minimal partition in which
the utterance is licensed. The standard implicatures then follow from a preference for
types with expert speakers. They are suspended if the utterance can only be optimal if
the expert assumption is violated. We show that this model can explain the examples in
(2).
2
The Standard Theory
In the standard theory (Gazdar 1979; Levinson 1983), suspension is explained by a conflict between scalar implicatures and clausal implicatures, which is resolved in favour of
the latter. Clausal implicatures are a special case of quantity implicatures. Like scalar
implicatures they result from a scale. They are not triggered by the occurrence of a weaker
expression within a proposition, but by the use of a weaker operator which subordinates
a proposition. The following table shows some examples:
a) stronger form b) weaker form c) implicature of weaker form
know A
believe A
3A ∧ 3¬A
necessarily A
possibly A
3A ∧ 3¬A
A and B
A or B
3A ∧ 3¬A ∧ 3B ∧ 3¬B
From the assumption that the speaker follows the maxim of quantity, it follows that the
use of the weaker form believe A implies that the speaker does not know A, hence, for
all what the speaker knows, it is possible that A is false (3¬A). Applied to our example
(1b), it follows that I believe that some of the boys came to the party implicates that 3
all came & 3 not all came. And analogously it follows that possibly all of the boys came
implicates that 3 all came & 3 not all came.
Scalar and clausal implicatures may get in conflict with each other. This conflict is
resolved by the above mentioned incremental process which amplifies utterance meaning
⊲ LoLa 10/Anton Benz: On the suspension of implicatures
74
(Gazdar 1979). The examples in (2) pose a principled problem for this account as know,
being the strongest expression, does not generate clausal implicatures, and hence should
not suspend any of the scalar implicatures.
Before turning to the critical examples, we first consider an explanation which derives
scalar implicatures and their suspension directly from the assumption that the speaker
follows the Gricean maxims. We write A(X) for the sentence frames X of the boys came to
the party’ and X students passed with open quantifier position X; A(∀) and A(∃) means
that all and some fill the open position, and A(∃ ∧ ¬∀) that some but not all occupies
it; UA means that the speaker uttered A; and 2A means that the speaker is convinced
that A. “(Quality)” says that the speaker can only utter what he believes to be true, and
“(Quantity)” that he will always utter the strongest form he is convinced of:
Derivation of Implicature in Example 2b Suspension of Implicature in Example (1b)
1. 2A(∀) → UA(∀) (Quantity)
2. UA(∃) (fact)
3. ¬2A(∀) (follows from lines 1 & 2)
4. 2A(∃) (follows from 2 and Quality)
5. 2A(¬∃) ∨ 2A(∃ ∧ ¬∀) ∨ 2A(∀) (Expert)
6. 2A(∃∧¬∀) (follows from lines 3., 4.,
and 5.)
1. 2A(∃) ∧ 3A(∀) (logical form of utterance and Quality)
2. 2A(¬∃) ∨ 2A(∃ ∧ ¬∀) ∨ 2A(∀) (Expert)
3. 2A(∀) (follows from previous lines)
4. 2A(∀) → UA(∀) (Quantity)
5. Contradiction (because speaker did
not utter A(∀))
6. ¬(Expert) ≡ 3¬A(¬∃) ∧ 3¬A(∃ ∧
¬∀) ∧ 3¬A(∀)
7. 2A(∃) ∧ 3A(¬∀) ∧ 3A(∀) (from the
first and the previous line)
These derivations are close to Grice’ (1989) original account of implicatures. The (Expert)
assumption says that the speaker knows whether A of none, some but not all or all is
true. The suspension examples with possibly make it very clear that this assumption is
necessary for deriving the standard scalar implicature. That it is often omitted indicates
that it is an assumption that is tacitly made about speakers in normal situations. In the
suspension example (1b), this assumption conflicts with the assumption that the speaker
follows (Quantity). This conflict makes it necessary for the interpreter to decide which
assumption to give up. With Grice, we assume that, in the face of a conflict, the interpreter
holds on to the assumption that the speaker follows the maxims. We can see from the
second derivation that giving up the (Expert) assumption not only suspends the scalar
implicature 2A(∃∧¬∀) but also explains the clausal implicature 3A(¬∀)∧3A(∀). In the
following section, we present a general framework that allows integrating this reasoning
in a game theoretic model of implicatures.
3
The Game Theoretic Model
Before introducing a general model, we first provide an analysis of example (2a). To
make the scenario more specific, we assume that the inquirer has to decide between two
candidates a and b. Let S(x) and P (x) be the propositions ‘x speaks Spanish’ and ‘x
speaks Portuguese’, and let 2X mean that the speaker is convinced that X. We represent
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75
the resulting 16 possible worlds on the left side of the table in figure (3). On the right
side of the top row, 8 answers with low complexity are listed.
(3) A model of example (2a)
S(a)
+
+
+
+
+
+
+
+
−
−
−
−
−
−
−
−
P (a)
+
+
+
+
−
−
−
−
+
+
+
+
−
−
−
−
S(b)
+
+
−
−
+
+
−
−
+
+
−
−
+
+
−
−
P (b)
+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
S(a)
1
1
1
1
1
1
1
1
·
·
·
·
·
·
·
·
P (a)
1
0
1
1
·
·
·
·
1
1
1
1
·
·
·
·
¬S(a)
·
·
·
·
·
·
·
·
1
1
1
1
1
1
1
1
¬P (a)
·
·
·
·
1
1
1
1
·
·
·
·
1
1
0
1
2S(a)
1
1
1
1
1
1
1
1
·
·
·
·
·
·
·
·
2P (a)
1
0
1
1
·
·
·
·
1
1
1
1
·
·
·
·
2¬S(a)
·
·
·
·
·
·
·
·
1
1
1
1
1
1
1
1
2¬P (a)
·
·
·
·
1
1
1
1
·
·
·
·
1
1
0
1
We assume that the inquirer knows the truth values of S(b) and P (b), and that the
question in (2a) is about a. Furthermore, we assume that a job candidate with knowledge
of Spanish is much preferred over a candidate without it. We interpret much preferred as
meaning that if the inquirer has to choose between a candidate x of whom he knows that
he speaks Spanish, and another candidate y of whom it is uncertain whether he speaks
Spanish, he will decide for the candidate x. If the job candidates’s knowledge of Spanish
is equal, then the candidate who speaks Portuguese is preferred. The numbers 1 and 0
in the answer columns mean that the inquirer will (1) or will not (0) choose an optimal
candidate given that he learns the answer and assumes all worlds to be equally probable.
The dots mean that the respective answer is not admissible. For example in the second
row, the inquirer knows that S(b) ∧ ¬P (b). If he learns that S(a), he will decide for
candidate a as a may also know Portuguese with probability 21 . This is the correct choice
(1). If he learns that P (b), then he will decide for b as he knows that b speaks Spanish,
and a Spanish speaking candidate is much preferred over a candidate who doesn’t speak
Spanish, which may be true of a with probability 21 . This is the wrong choice (0). In all
rows we find that the boxed answers lead to the same results as the unboxed answers.
What are the optimal answers? Lets first assume that the speaker is an expert about
a. He is not an expert about b, i.e., he knows the truth values of S(a) and P (a) but not of
S(b) and P (b). The optimal answers in situations in which S(a) ∧ P (a) holds are S(a) and
2S(a). We write Op++ = {S(a)} for the set of un-boxed optimal answers. Analogously,
we arrive at Op+− = {S(a), ¬P (a)}, Op−+ = {P (a), ¬S(a)}, and Op−− = {¬S(a)}.
Hence, from the fact that the speaker only produces optimal answers, the inquirer can
infer that answer P (a) implicates that ¬S(a), and that answer ¬P (a) implicates that
S(a). The optimal answers S(a) and ¬S(a) do not lead to additional implicatures.
Before considering the non-expert case, we present the game theoretic framework in
which this reasoning can be carried out.
Our basic structures representing utterance situations are interpreted support problems, i.e., tuples σ = hΩ, PS , PH , F, A, u, c, [[ . ]]i consisting of (1) a finite set of possible
worlds Ω, (2) probability distributions PS for the speaker and PH for the inquirer, or
hearer, (3) a finite set of forms F from which the speaker chooses his answer, (4) a set
A of hearer actions, (5) a joint payoff function u, (6) a function c measuring the complexity of forms, and (7) an interpretation function [[.]] which maps forms to subsets of
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Ω. If, in situation v, the speaker chooses form F and the hearer action a, then the joint
payoff is u(v, a) − c(F ). The costs c(F ) of forms are assumed to be nominal, i.e., positive but very small. We represent Grice’ maxim of quality by the assumption that the
speaker can only choose answers which he believes to be true, i.e., answers from the set
Admσ = {F ∈ F | PS ([[F ]]) = 1}. We further assume that the beliefs of the speaker
and the hearer cannot contradict each other, i.e., ∃v ∈ Ω : PS σ (v) > 0 ∧ PH σ (v) > 0.
For a full theory, we need a general method for finding Nash equilibria (S, H) of speaker
and hearer strategies which provide a solution to a given set of support problems S.
As argued for in Benz & van Rooij (2007), we assume that backward induction is used,
which replaces the maxim of quantity. This means that the hearer will choose his actions
P
from the set B(F ) of all actions a for which EU H (a|[[F ]]) = v PH (v|[[F ]]) u(v, a) is maximal. The resulting hearer strategy in σ is a probability distribution H σ ( . |F ) over B(F ).
If the speaker knows the state of the hearer, i.e. if he knows σ, then he will optimise
P
P
EU S (F ) = v PS (v) a H(a|F ) u(v, a).
In our model of example (2a), we find that the speaker is uncertain about the hearer
state. Let [σ]S be the subset of support problems which differ from σ only with respect
to PH . Then we can represent the speaker’s uncertainty by a probability distribution
µσ ( . |v) over [σ]S . It may depend on the state of the world v. In this case, the speaker
P
P
P
′
will optimise EU S σ (F ) = v PS (v) σ′ µσ (σ ′ |v) a H σ (a|F ) u(v, a). The optimal answers
Opσ are then the answers F ∈ Adm for which EU S σ (F ) becomes maximal. This has the
consequence that Opσ = Opσ′ for all σ ′ ∈ [σ]S . In e.g. Op+− , +− denoted the equivalence
class of all σ for which PS σ (S(a) ∧ ¬P (a)) = 1. In our model (3), we assumed that the
hearer knows the properties of b; this implies that the hearer state is determined by v and
that therefore µσ (σ ′ |v) = 1 or µσ (σ ′ |v) = 0.1
We can represent the maxim of manner by the assumption that the speaker chooses
optimal answers of minimal complexity. This defines optimal speaker strategies S( . |σ)
as probability distributions over Opσ . It follows that an utterance of a form F carries
the information that F ∈ Opσ . Hence, in σ, the hearer can infer from an utterance
′
′
of F that a proposition R holds if PS σ (R) = 1 for all σ ′ such that PH σ = PH σ and
F ∈ Opσ . In order to count as an implicature, this fact must also be common ground. Let
O(F ) = {σ | F ∈ Opσ } be the set of all support problems σ for which F is optimal. We
denote the common ground updated with the information that F is optimal by CGσ (F )
and define it as follows: Let [σ]S be as before, and let [σ]H be the analogue set of all
support problems which only differ from σ with respect to PS ; let E 0 σ := {σ} ∩ O(F ),
S
S
E n+1 σ := O(F ) ∩ {[σ ′ ]S , [σ ′ ]H | σ ′ ∈ E n σ }, and CGσ (F ) = n E n σ . We tacitly assumed
here that in all σ each interlocutor Y believes that all σ ′ ∈ [σ]Y are possible. In analogy
to logical notation, we write (S, σ) ||= F +> R if the utterance of F implicates that σ ∈ R
for sets of support problems S and R ⊆ S. This leads to the following definition:
(4) (S, σ) ||= F +> R :⇔ CGσ (F ) ⊆ R.
If we identify the meaning [[F ′ ]] of a form with the set RF = {σ | PS σ ([[F ′ ]]) = 1}, then
′
we arrive at the special case: (S, σ) ||= F +> F ′ :⇔ ∀σ ′ ∈ CGσ (F ) PS σ ([[F ′ ]]) = 1. If
(S, σ) ||= F +> R holds for all σ ∈ S, then we can write S ||= F +> R.
In order to account for the suspension of implicatures, we assume that for each form
F there is a set M (F ) which represents the normality assumptions about utterances of F .
The set M (F ) is derived from a preferential model hM, ≤i consisting of a partition M
of a given set S of support problems and a linear well-founded order ≤. The preferential
1
A more detailed model with uncertainty about hearer states was presented in Benz (2004).
⊲ LoLa 10/Anton Benz: On the suspension of implicatures
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model tells us which utterance situations are normal in general. For simplicity we assume
that the speaker knows to which M ∈ M the actual support problem σ belongs, i.e., we
assume σ ∈ M → [σ]S ⊆ M . We define M (F ) then as the ≤-minimal set M for which
an utterance of F is licensed. The remaining problem is then to spell out what exactly
licensed means. Before we address this problem, we adjust our definition of implicatures.
If M (F ) is given, we can, in analogy to preferential nonmonotonic logics, characterise the
preferred implicatures by:
(5) hM, ≤i ||≈ F +> R :⇔ M (F ) ||= F +> R.
Our final task is then to find a suitable definition of M (F ). In case of the model
in (3), we can assume that M consists of the set M of support problems in which the
speaker is an expert, and the set M of support problems in which the speaker is not an
expert. The normal case is the expert case. From (5) it follows that it always holds that
F +> M (F ). This entails that, if the speaker uses a form F which is optimal in a most
normal situation, the addressee will infer that the situation is indeed most normal. Hence,
if the speaker uses F in a non–normal situation, it will mislead the hearer if nothing stops
him from thinking that the situation is normal. Let M σ be the unique M ∈ M for which
σ ∈ M . Then, the speaker’s goal of avoiding misleading forms will lead him to restrict
his choice to forms F which are not elements of any Opσ′ for which M σ′ < M σ . This
condition must be added to the definition of admissible answers Admσ . This addition
entails that, in non–normal support problems, Opσ will now contain other forms than
before. This leads to the following definition of normal models for utterances of F :
(6) M (F ) = min{M ∈ M | O(F ) ∩ M 6= ∅}
We finally show how this can be applied to the model in (3). We say that the speaker
is an expert about a if (2S(a) ∨ 2¬S(a)) ∧ (2P (a) ∨ 2¬P (a)); hence, he is a non-expert
iff (3¬S(a) ∧ 3S(a)) ∨ (3¬P (a) ∧ 3P (a)). As 2φ is equivalent to PS (φ) = 1, uttering,
e.g., 2P (a) entails in conjunction with the assumption that the speaker is not an expert
that PS (S(a)) < 1 ∧ PS (¬S(a)) < 1, i.e., 3¬S(a) ∧ 3S(a). Hence, M ||= 2P (a) + >
(3¬S(a) ∧ 3S(a)). As we have seen before, S(a), P (a), ¬S(a), and ¬P (a) are all optimal
answers for some support problem in which the speaker is an expert, but none of the boxed
formulas 2S(a), 2P (a), 2¬S(a), and 2¬P (a) is one. Hence, the un-boxed formulas are
not admissible in non-expert situations, but the boxed formulas are. As by assumption the
hearer solves his decision problem by backward induction in which he only takes the literal
meaning of a form into account, it follows that the more complex forms 2P (a) ∧ 3S(a)
or 2P (a) ∧ 3¬S(a) do not increase the speaker’s expected utility. Hence, in all situations
σ ∈ M in which the speaker knows that P (a) but is not an expert, 2P (a) is an optimal
answer, and it implicates that the speaker does not know whether S(a). This explains
the critical example (2a). We also find that the utterance 2¬P (a) implicates that the
speaker does not know whether S(a), as do the answers P (a) ∧ 3S(a), P (a) ∧ 3¬S(a),
¬P (a) ∧ 3S(a), and ¬P (a) ∧ 3¬S(a).
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